Passing through the waveguide, the input signal \(\alpha e^{-i\omega t}\)  interacts resonantly with the qubit inside, changing both its population and coherence. Due to the presence of the drive and qubit internal losses, which are not included in the Wigner-Weisskopf theory, the system is better described using the input-output formalism [1]}. Being an open quantum system the qubit state in the rotating frame follows a spiral trajectory inside the Bloch sphere during the pulse, as shown in figure REF (a). Recording the output signal \(\langle a_{out}(t) \rangle \)  after the drive is abruptly stopped and fitting it with an input-output theoretical model (see Appendix ) allows to effectively measure the qubit coherence \(\langle \sigma ^{-} \rangle \)  as a function of time. Having access to \(\langle \sigma ^{-}(t) \rangle \)  during the drive pulse as well as the decay and assuming that the qubit is initially in the ground state \({g}\) , one can access the qubit population \(\langle \sigma ^{z}(t) \rangle \)  as well.
This information is enough for a qubit state tomography on an open quantum system. Similar protocols have already been implemented in circuit QED systems [2]}, [3]}, [4]}.
